Solve for x: x^4 + x^3 + x^2 + x + 1 = 0.
If anyone's curious, the four complex roots are the following:
x ≈ 0.309 + 0.951i
x ≈ 0.309 - 0.951i
x ≈ -0.809 + 0.588i
x ≈ -0.809 - 0.588i
(using a computer, of course )
Polar form:
$\displaystyle x = \cos \frac{2\pi}{5} + i\sin \frac{2\pi}{5}$
$\displaystyle x = \cos \frac{8\pi}{5} + i\sin \frac{8\pi}{5}$
$\displaystyle x = \cos \frac{4\pi}{5} + i\sin \frac{4\pi}{5}$
$\displaystyle x = \cos \frac{6\pi}{5} + i\sin \frac{6\pi}{5}$
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I think there's a typo in pacman's last post. Instead of
$\displaystyle \left(x + \frac{1}{x}\right)^2 + \left(x + \frac{1}{x}\right) + 1 = 0$
it should have been
$\displaystyle \left(x + \frac{1}{x}\right)^2 + \left(x + \frac{1}{x}\right) {\color{red}-}\; 1 = 0$
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